The Astrophysical Journal, 793:112 (11pp), 2014 October 1 doi:10.1088/0004-637X/793/2/112C© 2014. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

NONLINEAR FORCE-FREE FIELD MODELING OF THE SOLAR MAGNETIC CARPETAND COMPARISON WITH SDO/HMI AND SUNRISE/IMAX OBSERVATIONS

L. P. Chitta1,2, R. Kariyappa1, A. A. van Ballegooijen2, E. E. DeLuca2, and S. K. Solanki3,41 Indian Institute of Astrophysics, Bangalore 560 034, India

2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street MS-58, Cambridge, MA 02138, USA3 Max-Planck-Institut fur Sonnensystemforschung, Justus-von-Liebig-Weg 3, D-37077 Gottingen, Germany

4 School of Space Research, Kyung Hee University, Yongin, Gyeonggi-Do 446-701, KoreaReceived 2014 April 25; accepted 2014 August 1; published 2014 September 12

ABSTRACT

In the quiet solar photosphere, the mixed polarity fields form a magnetic carpet that continuously evolves due todynamical interaction between the convective motions and magnetic field. This interplay is a viable source to heatthe solar atmosphere. In this work, we used the line-of-sight (LOS) magnetograms obtained from the Helioseismicand Magnetic Imager on the Solar Dynamics Observatory, and the Imaging Magnetograph eXperiment instrumenton the Sunrise balloon-borne observatory, as time-dependent lower boundary conditions, to study the evolution ofthe coronal magnetic field. We use a magneto-frictional relaxation method, including hyperdiffusion, to produce atime series of three-dimensional nonlinear force-free fields from a sequence of photospheric LOS magnetograms.Vertical flows are added up to a height of 0.7 Mm in the modeling to simulate the non-force-freeness at thephotosphere–chromosphere layers. Among the derived quantities, we study the spatial and temporal variations ofthe energy dissipation rate and energy flux. Our results show that the energy deposited in the solar atmosphere isconcentrated within 2 Mm of the photosphere and there is not sufficient energy flux at the base of the corona to coverradiative and conductive losses. Possible reasons and implications are discussed. Better observational constraints ofthe magnetic field in the chromosphere are crucial to understand the role of the magnetic carpet in coronal heating.

Key words: Sun: atmosphere – Sun: corona – Sun: magnetic fields – Sun: photosphere

Online-only material: color figures

1. INTRODUCTION

The dynamical evolution of magnetic field in the solar photo-sphere holds the key to solving the problem of coronal heating.High-resolution observations show that a mixed polarity field,called the magnetic carpet (Schrijver et al. 1997), is spreadthroughout the solar surface. The loops connecting these ele-ments pierce through the atmosphere before closing down atthe photosphere. The random motions of these small elementscaused by relentless convective motions in the photosphere area favorite candidate to explain the energy balance in the solaratmosphere (for example, Schrijver et al. 1998; Gudiksen &Nordlund 2002; Priest et al. 2002). In three-dimensional (3D)magnetohydrodynamic (MHD) models, small-scale footpointmotions drive dissipative Alfven wave turbulence in coronalloops (van Ballegooijen et al. 2011). Also, the convective mo-tions promote magnetic reconnection and flux cancellation—viable mechanisms to heat the corona5 (for example, Longcope& Kankelborg 1999; Galsgaard & Parnell 2005).

The magnetic carpet typically contains magnetic features withmagnetic flux ranging from 1016–1019 Mx, a part of which arekilo Gauss flux tubes, also in the internetwork (Lagg et al.2010). The carpet is continually recycled with newly emergingflux replacing the pre-existing flux. Magnetic elements split,merge, and cancel due to granular action (Iida et al. 2012; Yanget al. 2012; Lamb et al. 2013). Additionally, recent observationsshow small-scale swirl events in the chromosphere, possibly dueto the rotation of photospheric elements (Wedemeyer-Bohm& Rouppe van der Voort 2009). Also, the magnetic elementsdisplay significant horizontal motions that can reach supersonic

5 The evolution of the magnetic carpet is also studied in the context ofacceleration of slow and fast solar wind (Cranmer & van Ballegooijen 2010;Cranmer et al. 2013).

speeds (e.g., Jafarzadeh et al. 2013). All these dynamical aspectsof the magnetic carpet make the overlying field non-potential—asource of magnetic energy (for a review on small-scale magneticfields, see de Wijn et al. 2009). There is indirect observationalevidence for the existence of non-potential structures in the quietSun (Chesny et al. 2013).

The vector magnetic field (B) or its line-of-sight (LOS)component (Bz) at the photosphere is used to infer coronalfields due to the unavailability of their direct measurementsin corona. Earlier studies of the co-evolution of magneticcarpet and coronal field were mainly through the potential field(current-free) extrapolations of the photospheric magnetic field(e.g., Close et al. 2004; Schrijver & van Ballegooijen 2005).Recently, Meyer et al. (2013) have used nonlinear force-freefield extrapolations of the observed Bz to study the magneticenergy storage and dissipation in the quiet Sun corona. Theyconcluded that the magnetic free energy stored in the coronalfield is sufficient to explain structures like X-ray bright points(XBP) and other impulsive events at small scales (for reviews onthe force-free magnetic fields, see Schrijver et al. 2006; Metcalfet al. 2008; Wiegelmann & Sakurai 2012). Wiegelmann et al.(2013) have used a 22 minute time sequence of very high-resolution vector magnetograms to extrapolate the field intothe upper atmosphere under the potential field assumption, andargued that the energy release through magnetic reconnection isnot likely to be the primary contributor to the heating of solarchromosphere and corona in the quiet Sun. To test the basis ofmagnetic reconnection between open and closed flux tubes asa plasma injection mechanism into the solar wind, Cranmer &van Ballegooijen (2010) used Monte Carlo simulations of themagnetic carpet. They also concluded that the slow or fast solarwind is unlikely to be driven by loop-opening processes throughreconnections. The works of Cranmer & van Ballegooijen

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Figure 1. Contextual figure showing the photospheric LOS magnetograms taken from the respective time sequences: (a) SDO/HMI, (b) Sunrise/IMaX, and (c) modelmagnetogram from the magnetic sources (see the text for details). Note the larger FOV of SDO/HMI compared to the Sunrise/IMaX and Source Model. A white boxshown in panel (a) is approximately the size of the FOVs in panels (b) and (c). To facilitate the display, all the magnetograms are saturated at ±50 G.

(2010; using models to study solar wind acceleration) andWiegelmann et al. (2013; using observations to study solaratmospheric heating) arrive at similar conclusions, disfavoringa significant role of the evolution of the magnetic carpet insupplying energy to the corona and solar wind, in contrast toMeyer et al. (2013).

A possible explanation for the above conflicting results is thatthe potential field studies simplify the magnetic topology and donot include dynamic aspects like currents and other nonlineareffects. Also, as the methods of analysis are not the same, itmay not be so straightforward to compare the results from thosestudies. With an ever increasing quality of the observationsshowing more intermittent flux filling the solar surface, it isvalid to inquire about its contribution to the dynamics of theupper solar atmosphere and advance our knowledge toward aholistic picture of the magnetic connection from photosphere tocorona, particularly in the quiet Sun.

To rigorously address these issues, a continuous monitoringof the Sun’s magnetic field is desired. This valuable facilityis provided by the Helioseismic and Magnetic Imager (HMI;Scherrer et al. 2012) on board the Solar Dynamics Observatory(SDO; Pesnell et al. 2012). SDO/HMI obtains full-disk magne-tograms of the Sun at 0.′′5 pixel−1 with 45 s cadence. To probe themagnetic field at an even higher spatial resolution, the ImagingMagnetograph eXperiment (IMaX; Martınez Pillet et al. 2011)instrument on the Sunrise balloon-borne observatory (Solankiet al. 2010; Barthol et al. 2011) recorded 33 s cadence obser-vations at 0.′′055 pixel−1. To better understand the role of themagnetic carpet, we use the SDO/HMI and the Sunrise/IMaXLOS magnetic field observations as lower boundary conditionsin a time-dependent nonlinear force-free modeling of the coro-nal field and report our comparative findings. The rest of thepaper is structured as follows. In Section 2, we describe the datasets and model setup. In Section 3, the main results of the workare presented. We conclude in Section 4 with a summary, andimplications of the results are discussed.

2. OBSERVATIONS AND MODEL SETUP

In this work, we used a one day long time sequence of theSDO/HMI LOS magnetic field observations at the disk centerand compared the results with the higher-resolution observationsat the disk center obtained from the Sunrise/IMaX instrument.In Section 2.1, we briefly describe the data sets and Section 2.2

deals with the setup of the simulation for the magnetic fieldextrapolations.

2.1. Data Set

HMI Data (Set 1). This set consists of a tracked, one daylong time sequence of the LOS magnetograms observed at thedisk center on 2011 January 08, starting at 00:00 UT. With afield of view (FOV) of 187 × 187 Mm2 and a time cadence of45 s, these observations cover several magnetic network patchesin the quiet Sun. In Figure 1(a), we show a snapshot from theHMI observations. The magnetic flux density is saturated at50 G. The observations are found to be closely in flux balanceover the entire duration. To retain the weaker field at the lowerboundary, no smoothing is applied to the data. The larger regionconsidered here allows us to compare various results betweensub-regions with varied magnetic configurations. The details aregiven in Section 3.

IMaX Data (Set 2). These level 2 data correspond to imag-ing spectropolarimetric observations of Fe i at 5250.2 Å. Theobservations were recorded at the disk center on 2009 June 09,starting at 01:30 UT. The spatial resolution of these observationsis 10 times better than the HMI sequence. On the downside,IMaX observations span over time period of only ≈30 minutes.From the original frames, the degraded edges because of theapodization (required by the phase diversity restoration tech-nique) during the reduction process have been discarded. Weextracted a 30 × 30 Mm2 region for further analysis. The LOSmagnetograms are derived by taking the ratio of Stokes V andI with a calibration constant (see Equation (17), Martınez Pilletet al. 2011). The data are binned to a pixel scale of 119.2 km(Figure 1(b)). This effectively reduces the noise in the data by upto a factor of three, but it also reduces signal due to small-scalemixed polarity fields. Similar to the HMI data, IMaX data arealso found to be closely in flux balance. Due to the limitationsof the time duration of the data set, the evolution of the coronalmagnetic field cannot be studied for a longer period. For thispurpose, we have constructed a series of artificial LOS magne-tograms named Source Model. The details of the Source Modelare given below.

Source Model (Set 3). The Source Model is made to comple-ment the IMaX observations, but for a longer duration of time(see the Appendix for details). The model is initiated with 50magnetic sources placed randomly on a mesh of hexagonal grids

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Figure 2. Mean magnetic field as a function of time for the three magnetogram types: (a) SDO/HMI (solid line, the mean magnetic field is derived from a regionmarked with the white box in Figure 1(a)), Source Model (dashed line), and (b) Sunrise/IMaX. The y axis scaling for both panels is the same.

(which has the dimensions of IMaX FOV considered above).The mesh is periodic in x and y directions. The sources includeboth positive and negative polarity elements such that the netflux is zero. The sources are allowed to move along the edgesof each hexagonal grid, which has a length unit of ≈1 Mm. Allthe sources move with a uniform velocity of 1.5 km s−1, whichis in the range of typical observed velocities of the small-scalemagnetic elements due to the interactions with granules (fore.g., Chitta et al. 2012, and references therein).

During the time evolution, the flux emergence, cancellation,along with splitting and merging of the elements are the possibleways of interactions among the magnetic elements at the cornersof each grid. The LOS magnetic field reached a quasi-statisticalequilibrium almost 15 hr after the initiation. During this time,the total magnetic flux increased from 1019 Mx to 1020 Mx. Outof the total 48 hr of time evolution, which includes both the risetime and equilibrium period, a portion of 8.5 hr time sequenceis used for further analysis. The total flux in the Source ModelFOV roughly matches with that of the IMaX observations. InFigure 1(c), we show a snapshot of the LOS magnetogram fromthe Source Model.

To ensure ∇ · B = 0, fractional flux imbalance in the observeddata (HMI and IMaX) is corrected by dividing the positive fluxwith an absolute ratio of integrated positive flux to the negativeflux in the FOV. This method is justified only if the ratio is closeto unity to begin with, in other words, the observations must beclosely in flux balance. This is the case in both the observationswe used in this study. In Figure 2, we plot the mean magneticfield (〈|Bz|〉) as a function of time for all the data sets. In theleft panel, the solid line is for the HMI set and dashed line isfor the Source Model. The right panel shows the IMaX meanmagnetic field. We emphasize that the IMaX LOS magnetic fieldis derived from a simple ratio method, which underestimates theflux density in stronger elements (Martınez Pillet et al. 2011).

2.2. Simulation Setup

In this section, we describe the simulation setup and theequations we solve to derive the 3D magnetic field above thephotosphere.

For Set 1, the computational box is a 512 × 512 × 250 cellvolume covering 187 × 187 × 91.4 Mm3 in physical space.For Sets 2 and 3, the computational domain is much smaller,covering only 30.5 × 30.5 × 18 Mm3 with 256 × 256 × 150

cells. The simulation box is periodic in the x and y directionsand closed at the top. LOS magnetograms of Sets 1–3 are therespective boundary conditions in the xy plane at z = 0. Initially,at time t = 0, a potential field is assumed to fill the box, whichis then evolved in time into nonlinear force-free states with theevolving boundary conditions.

The rate of change of vector potential A is related to themagnetic field B through the induction equation

∂A∂t

= v × B + ε, (1)

where ε is the hyperdiffusion, defined as

ε = BB2

∇ · (η4B2∇α). (2)

α = j · B/B2 is the force-free parameter and η4 is the hyperdiffu-sivity. Some form of magnetic diffusion is needed in any numer-ical MHD simulation to suppress the development of numericalartifacts on small spatial scales. Here we use hyperdiffusion (in-stead of ordinary resistivity) to minimize the direct effect of thediffusion on larger scales. van Ballegooijen & Cranmer (2008)presented a theory of coronal heating that draws energy withhyperdiffusion from the dissipation of nonpotential magneticfield. It conserves the mean magnetic helicity and smooths thegradients in α (Boozer 1986; Bhattacharjee & Hameiri 1986).It has been proposed that tearing modes can promote turbu-lence in 3D sheared magnetic field, which causes hyperdiffusion(Strauss 1988).

Equation (1) is evolved using a magneto-frictional relaxationtechnique (Yang et al. 1986), which assumes that the plasmavelocity (v, in this case, the magneto-frictional velocity) isproportional to the Lorentz force (j × B), given by

v = 1

ν

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, (3)

where ν is the frictional coefficient. The numerical values ofη4 and ν−1 (tabulated in Table 1) are determined in part bythe requirement of numerical stability of the code, and by thelength (L) and time (T) units of the simulations. Mackay et al.(2011) and Cheung & DeRosa (2012) used magneto-frictionalrelaxation technique to study the time evolution of active regionson much larger scales.

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Figure 3. Top: volume rendering of the extrapolated Bz using SDO/HMI observations as lower boundary conditions. Shown here is the time evolution of Bz of anephemeral region, with consecutive frames lying two hours apart, in a box of ≈55 × 55 × 14 Mm3 volume, with its bottom surface centered at (70 Mm, 90 Mm)of Figure 1(a). The red (blue) colored areas are the regions of negative (positive) polarity. Bottom: the distribution of Bz (saturated at ±50 G) in the photospherecorresponding to the respective panels in the top segment is shown.

(A color version of this figure is available in the online journal.)

Table 1Values of L, T, η4, and ν−1 Used in the Magnetic Modeling

L T η4 ν−1

(Mm) (s) (km4 s−1) (km2 s−1)

HMI 3.12 900 1.05 × 107 1 × 103

IMaX 1.02 165 6.48 × 106 0.6 × 103

Source Model 1.02 680 1.57 × 106 0.1 × 103

It is known that the magnetic field in the photosphere isnon-force-free due to the high plasma β. Metcalf et al. (1995)calculated the Lorentz force of an active region as a functionof height. They concluded that the field becomes force-free

higher in the atmosphere above approximately 400 km. Themixed polarity fields in the magnetic carpet are weaker andmay be dominated by gas pressure even in the chromosphereas suggested by the simulations of Abbett (2007), which,however, do not include a realistic treatment of radiative transferfor the photospheric and chromospheric layers. To mimic theadditional forces on the magnetic field in the photosphereand low chromosphere, we use the method first described byMetcalf et al. (2008), where a second term is added to themagnetofrictional velocity

v =(

1

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)× B/B2. (4)

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The additional term is present only at low heights (up to 0.7 Mm)and is the projection of a constant upward velocity (v1z) ontothe plane perpendicular to the local magnetic field. The quantityv1 is constant over horizontal planes and independent of themagnetic field. This new force marginally prevents the field fromsplaying out at the lower boundary. In the present paper, we usev1 = 1.5 km s−1 for IMaX/Source Model, and v1 = 3.4 km s−1

for HMI data set. This has only a minor effect on the fluxconcentrations in our model. Similar to η4 and ν−1, the choiceof v1 is defined by the L and T units of the respective models.

3. RESULTS

In the top segment of Figure 3, we present volume renderingof Bz in a sub-volume from HMI simulations at four instances,separated by two hours each. At the lower boundary of thissub-volume, an emerging bipole is seen, with stronger negativepolarity (centered on the black square, Figure 1(a)). The red(blue) coloration covers regions of negative (positive) polarity.As the bipole emerges into the atmosphere, pre-existing coronalfield responds to it, and eventually becomes bipolar in time. Notethat this is a 3D rendering. To see through the cube, we usedincreasing transparency of layers with height (so that we can seetill the bottom through the layers above). Also, to account forthe decrease in field strength with height, we scaled each layer(at a given height) in respective panels separately. In the bottomsegment, we show Bz at photospheric level corresponding to therespective panels in the top segment. The purpose of this figureis to show how the coronal field responds to an emerging bipolein the photosphere.

The magnetic free energy (Efree), which is an excess of non-potential magnetic energy (in this case, the nonlinear force-freeenergy) over the potential magnetic energy is calculated. Ouraim is to emphasize the height dependence of the quantitiesrelevant to the energy budget of the solar corona. To this end,we calculate Efree as a function of height, given by

Efree(z) = 1

8πΔz

∫∫ [B2

np(x, y, z) − B2p (x, y, z)

]dxdy, (5)

where Δz is the pixel length. B2np/8π and B2

p/8π are the non-potential, and potential magnetic energy densities, respectively.In the top panel of Figure 4, we plot the HMI Efree(z) (Δz =0.36 Mm). Integration is over the surface enclosed by the blacksquare in Figure 1(a). The curves represent the evolution ofan emerging bipole at four times (shown in Figure 3). It isobserved that most of Efree(z) (5–10×1025 erg) is available atheights below 5 Mm. These results are consistent with the freeenergy values reported in Meyer et al. (2013).

To estimate the non-potentiality of the magnetic energy, wecalculate a ratio γ (z), defined as

γ (z) =∫∫

B2np(x, y, z)dxdy∫∫

B2p (x, y, z)dxdy

. (6)

It is observed that the magnetic energy is close to the energyof the potential magnetic field, with an excess <20%. In thebottom panel of Figure 4, we plot γ (z) for the same time stepsas shown in the top panel. There is an apparent rise in thenon-potentiality with time, indicating the interaction of a newlyemerging flux with the pre-existing coronal field. However, bycombining the values of γ (z) and Efree(z), it can be seen thatthe coronal field over the HMI FOV is potential. We note thatin the bottom panel, γ (z) < 1 for some points in time above a

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Figure 4. Top: free energy of an emerging bipole located within the blacksquare in Figure 1(a). Each of the four curves plotted in a different line style isfor a different point in time, corresponding to the times of the snapshots plottedin Figure 3. The curves are labelled in the upper panel. At all times, Efree(z)falls rapidly with z, peaking below 5 Mm. Bottom: the ratio of non-potentialmagnetic energy to the potential magnetic energy (γ ) is plotted as a function ofheight (for the same time steps as shown in the top panel). The peaks of γ (z)shift toward higher z with time, indicating the interaction of the newly emergingfield with the pre-existing coronal field.

height of 10 Mm, which means a negative free energy. This isdue to the fact that the surface integration is performed only overa sub-region within the full FOV. The total volume integratedEfree, however, remains positive.

Magnetic free energy is necessary but not sufficient tocompletely quantify the energy supply to solar corona. Itis important to calculate the energy dissipation and energyflux, which can then be directly compared with the observedradiative and conductive losses. Earlier studies by Withbroe &Noyes (1977), Withbroe (1988), and Habbal & Grace (1991)placed observational constraints on the energy flux through thecoronal base, 105–106 erg cm−2 s−1 for a quiet Sun region.With magneto-frictional relaxation including hyperdiffusion,the energy dissipation per unit volume (Q) is calculated as

Q = B2

4π(ν|v|2 + η4|∇α|2), (7)

(see, Yang et al. 1986; van Ballegooijen & Cranmer 2008).The first and second parts on the right hand side of theEquation (7) are due to magneto-friction and hyperdiffusion,respectively. To understand the evolution of Q with respect

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Figure 5. HMI magnetic field configuration at the lower boundary ((a) and (e) saturated at ±50 G) and the energy dissipation (Q) at three layers ((b) and (f) 2.2 Mm;(c) and (g) 4 Mm; (d) and (h) 6.2 Mm) are shown. Note the change of grayscale ranges in the energy dissipation maps. The top and the bottom maps are separatedby 15 minutes in time and are rendered according to the same grayscales (for a given height). Though the magnetic field configuration has little apparent change, thelocations of the energy dissipation have changed significantly, suggesting a highly non-localized and intermittent nature of the time evolution of the magneto-frictionalrelaxation method.

to height at the timescale of minutes, we consider snapshotsof Q from HMI simulations, separated by 15 minutes. Theresults are shown in Figure 5. The top row shows Bz (lowerboundary, (a)) and Q at three heights ((b): 2.2 Mm, (c): 4 Mm,and (d): 6.2 Mm) at 5.25 hrs into the simulation. The bottomrow is same as the top row, but 15 minutes after panels(a)–(d). With very little change in the Bz over the duration,apparently, there is a drastic morphological change in Q (panels(b) and (f)). This is very interesting to note, suggesting thatthe energy dissipation process due to magneto-friction andhyperdiffusion is non-localized in time (i.e., the locations ofenergy dissipation rapidly change with time). This in turnmeans that the time evolution of spatially averaged Q at anytwo locations separated by some distance (having a differentunderlying magnetic morphology) would be similar. In otherwords, the average dissipation of magnetic energy is veryuniform in our simulations. However, the magnitude of Q has astrong height dependence (as seen in the grayscales of the rightthree panels). This was also indicated by Meyer et al. (2013).These large-scale changes in the atmosphere are caused bysmall-scale random footpoint motions of magnetic elements atthe photosphere.

To further quantify the above statements, Figure 6 plots thehorizontally averaged dissipation rate as a function of height andtime. The top left panel is Q plotted as a function of height fromthe HMI simulations. The two shaded bands are for two sub-regions of the HMI FOV (red, Figure 1(a) black square; blue,Figure 1(a) white square). The black square covers an ephemeralregion (ER) and the white square covers a less active quiet region(QR). At a given height, the width of each shaded band representthe range of minimum to maximum energy dissipated at thatheight. A vertical shaded rectangle is drawn to indicate the baseof the transition region and corona. Q monotonically decreaseswith height in a similar way for regions with completely differentunderlying Bz. In the top right panel, results from the SourceModel (red) and IMaX (blue) are shown for comparison withthose of HMI. It is noted that the fall of QSource Model/IMaX issteeper than QHMI.

Furthermore, we plot the averaged Q in two height ranges(0–2; 2–4 Mm) and study its time evolution. In Figure 6(c),we plot these results for the HMI simulations (red, ER; blue,QR; solid, 0–2 Mm; dashed, 2–4 Mm). These plots show that onaverage, the energy dissipation is uniform over the entire FOVand also fairly constant throughout the time evolution. Anotherfact to notice is that Q reaches its statistically mean value veryearly in the simulations. Recalling that the initial condition ofthe simulation is a potential field, the build-up of magneticshear of coronal field, and in turn the energy dissipationthrough hyperdiffusion is contributing only a fraction of totalQ. Magneto-friction acts as a dominant energy source in oursimulations. Figure 6(d) shows the evolution of QSource Model.

As a next step, we calculate the energy flux (F) through agiven horizontal surface. In a quasi-stationary state, the energyflux F (z) through height z is equal to the integral of Q(z) overall heights above z, given by

F (z) =∫ zmax

z

Q(z)dz, (8)

where zmax is the top of the simulation domain. The horizontalaverage of the calculated F (z) is plotted in Figure 7. Similarto Figure 6(a), in the left panel we show 〈F (z)〉 for two sub-regions within HMI FOV. 〈F (z)〉 of IMaX/Source Model areplotted in the right panel. All sets have a flux of ≈106 erg cm−2

s−1 at the photosphere. Almost all of this flux is dissipatedat heights below 2 Mm, i.e., in the chromosphere (for which,however, our model may not be valid since the field below1000–1500 km is not force-free). Over a quiet Sun, the coronalbase is typically above 3 Mm (see Figure 3, Withbroe & Noyes1977). At this layer, 〈FHMI(z)〉 ≈ 5 × 104 erg cm−2 s−1, and〈FIMaX(z)〉 ≈ 2×103 erg cm−2 s−1. These values are lower thanthe required flux in the quiet Sun corona.

To summarize Figures 6 and 7, three sets of magnetogramsused in this work show a similar trend. Close to the lowerboundary both Q(z), and F (z), respectively, are similar forall the cases. F (z) is > 105 erg cm−2 s−1 and appears to be

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Figure 6. Horizontal mean of energy dissipation plotted as a function of height (top panels) and time (bottom panels). (a) Mean energy dissipation over an ER (red)and a QR (blue) observed by HMI. (b) Energy dissipation in the IMaX data set (blue) and Source Model (red) averaged over the entire FOV. As an illustration, a graystriped box is drawn at the 2–6 Mm range to compare the energy dissipations between panels (a) and (b) at those heights. (c) Mean energy dissipation plotted vs. time,averaged over the height ranges 0–2 Mm (solid) and 2–4 Mm (dashed), respectively. (d) Same as (c) but for Source Model.

(A color version of this figure is available in the online journal.)

comparable with the coronal energy budget of the quiet Sun.This condition is no more satisfied for z >2–3 Mm. On average,there is an order of magnitude or more deficit in the requiredenergy flux to support the observed coronal energy losses. Thedeficit in F (z) is very noticeable in the higher resolution butsmaller FOV IMaX/Source Model cases.

Next, choosing the IMaX set as an example, we studied thedependence and sensitivity of the calculated energy dissipationand flux on the simulation parameters (ν−1, η4). As described inthe previous section, the numerical values of these parametersare constrained by the requirements that the code be numericallystable and that numerical artifacts are suppressed. The frictionalcoefficient and hyperdiffusivity can be written as ν−1 = θL2/Tand η4 = φL4/T , where θ and φ are dimensionless parameters,and L and T are the length and time units for each simulation.For the case of IMaX (and also for the Source Model), θ = 0.1and φ = 0.001; the corresponding values of ν−1 and η4 arelisted in Table 1. Numerical stability dictates that θ and φcannot be arbitrarily large and to suppress numerical artifacts,φ cannot be arbitrarily small. To test how lower values of these

parameters affect the final results, we considered three sub-cases: (a) θ = 0.01, φ = 0.0001; (b) θ = 0.01, φ = 0.001; and(c) θ = 0.1, φ = 0.0001. Furthermore, the energy flux with therespective new set of parameters is calculated and we denote itas F ′

θ,φ(z). The ratio F ′θ,φ(z)/F (z) is plotted in Figure 8 (solid

line, case (a); dashed line, case (b); dash-dotted line, case(c)).Here, F (z) is calculated with θ = 0.1 and φ = 0.001 (see theblue shaded band in the right panel of Figure 7). These resultsindicate that the energy dissipation and flux are sensitive to thefrictional coefficient, which can also be argued on the basis thatthe magneto-frictional energy is dominant in the simulations.

4. SUMMARY AND DISCUSSION

With an aim to understand the role of the magnetic carpetin the heating of the solar corona, we simulated the coronalmagnetic field using the disk center observations of quiet SunLOS magnetic field obtained from the SDO/HMI and Sunrise/IMaX. To overcome the limitations of short duration IMaXdata, we created a time sequence of synthetic magnetograms

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0 2 4 6 8 10 12 14Height (Mm)

100

101

102

103

104

105

106

107

Ene

rgy

Flu

x (e

rg c

m-2 s

-1)

HMI ERHMI QR

0 2 4 6 8 10 12 14Height (Mm)

Source ModelIMaX

Figure 7. Horizontal mean of the energy flux plotted as a function of height. The color schemes and legends in the left and right panels are the same as in Figures 6(a)and (b), respectively.

(A color version of this figure is available in the online journal.)

Figure 8. Dependence of energy flux on the dimensionless parameters (θ , φ) isshown for the case of the IMaX data set. The ratio F ′

θ,φ (z)/F (z) is plotted as afunction of height for three sets of parameters as marked in the plot legend (seethe text for details).

that roughly match the IMaX set in terms of the absoluteflux (Figures 1 and 2). A time series of the 3D nonlinearforce-free magnetic field is constructed with the magnetogramsas lower boundary conditions and potential fields as initialconditions. The coronal field is evolved using Equation (1)with a magneto-frictional relaxation technique (Equation (4)),including hyperdiffusion (Equation (2)). We incorporated thenon-force-free nature of the lower solar atmosphere (�0.7 Mm)by adding vertical flows that prevent the field from splaying outat the lower boundary (Equation (4)). These gentle flows weaklyincreased the field strength of the magnetic elements.

Next, we calculated Efree(z), Q(z), and F (z) to quantitativelyestimate the magnetic free energy, its dissipation, and energyflux injected at the base of corona. It is found that the energydissipation in simulations with moderate resolution (HMI)and much higher resolution (IMaX/Source Model) is verysimilar close to the lower boundary. At low heights in thesemodels, most of the heating occurs in short, low-lying loopsthat originate from the mixed-polarity magnetic carpet. Theaverage length of these loops is only a small fraction of thecomputational domain. Furthermore, these loops are highly

intermittent, facilitating uniform energy dissipation over theentire surface. These loops can be formed by interactions ofnearby elements (e.g., Wiegelmann et al. 2010) and also fromthe newly emerging flux in the internetwork (Centeno et al.2007).

Despite their intrinsic differences, the IMaX and SourceModel simulations are strikingly similar. Moving away fromthe lower boundary, we see that FSource Model/IMaX decreases bymore than two orders of magnitude at z = 2 Mm. This leavesonly a few percent of F at the coronal base. The energy fluxinto the corona as predicted by our model is approximately104 erg cm−2 s−1 (see the right panel of Figure 7), which is aboutone order of magnitude smaller than the estimated radiativeand conductive losses from the quiet-Sun corona as derivedfrom EUV and X-ray observations (e.g., Withbroe & Noyes1977; Withbroe 1988; Habbal & Grace 1991). We concludethat the present model, which is based on evolving force-free magnetic fields and uses realistic photospheric boundaryconditions, cannot account for the observed coronal heating ofthe quiet Sun.

Wiegelmann et al. (2013) used similar IMaX data to createa set of potential field models of the evolving magnetic carpet.They also concluded that the magnetic energy release associatedwith reconnections is not likely to supply the required energy toheat the chromosphere and corona. Our estimate for the energyflux F into the corona is below the upper limit provided byWiegelmann et al. (2013), indicating that flux estimates basedon force-free-field modeling are consistent with those basedon potential field modeling. However, both types of estimatesare well below the values needed to heat the corona. Thissuggests that both potential-field and force-free-field modelsunderestimate the amount of energy injected into the coronadue to photospheric footpoint motions and other effects in themagnetic carpet. This is perhaps not surprising because thesemodels make many simplifying assumptions about the structureand dynamics of the solar atmosphere. For example, the modelsignore the details of the lower atmosphere, where the densitydecreases with height over many orders of magnitude. Also,the models assume that the coronal magnetic field evolvesquasi-statically in response to the footpoint motions, whereas

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0 2 4 6 8 10Time (min)

0.00.2

0.4

0.6

0.8

1.01.2

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netic

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x (1

018 M

x)(a)

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0 2 4 6 8 10 12Time (min)

0.0

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(b) 0.6 2.8 5.1

6.8 9.1 11.3

0 2 4 6 8 10Time (min)

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(c)

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0 2 4 6 8 10Time (min)

0.0

0.5

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1.5

(d) 0.6 2.3 4.5

5.7 7.4 9.1

Figure 9. Various interactions among magnetic elements in the Source Model are illustrated here: (a) flux emergence, (b) flux emergence followed by a partial fluxcancellation, (c) flux splitting, and (d) total flux cancellation. Each panel is comprised of two segments. The top segment is a time sequence of magnetic mapsdescribing the interaction. Each map in the respective top segment covers an area of ≈2.4 × 2.4 Mm2. The numbers in the maps indicate time, in minutes, elapsedsince an arbitrary start time. The bottom segment is a plot of the magnetic flux integrated over the area of the map as a function of time. The scale of y axis in thebottom segment is set at 1018 Mx.

in reality, the response may be more dynamic and significantlymore energy may be injected in the form of Alfven waves (e.g.,van Ballegooijen et al. 2014).

However, the deficit in F is smaller in the HMI set as comparedto those of other models. Assuming that this discrepancy ismainly due to the larger HMI volume,6 it means that thereis a contribution from the long range magnetic connections,often reaching coronal heights, that was not possible with theIMaX set. This was also cited by Wiegelmann et al. (2013)as a reason for their too-low upper limit. Future observationswith high spatial resolution and large FOVs are required to

6 6.6× more grid cells covering 192.4× larger volume in physical space ascompared to the IMaX/Source Model volumes.

check and justify the contribution from long range magneticinteractions. Also, IMaX and HMI observed mainly differentkinds of features, with very different lifetimes. In the case ofHMI, it is predominantly network magnetic features (includingERs) and some internetwork fields; for IMaX, it is dominated byinternetwork fields. The latter are more likely to be horizontaland do not reach high into the atmosphere. Additionally, IMaXobservations were carried out in 2009, at the depth of the lastminimum in a very quiet part of the Sun, while the HMI datawere obtained in 2011, i.e., when the Sun was well on its wayinto the new cycle.

A relatively large FHMI(z) can be considered as a basal fluxto heat a diffused coronal region. Compact features like XBPs

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have a mean lifetime of eight hours (Golub et al. 1974) andan average emission height of 8–12 Mm above the photosphere(Brajsa et al. 2004; Tian et al. 2007). Our model predicts a fluxtoo small to account for the energy losses at those height ranges.

Another important issue to scrutinize here is the lower bound-ary condition itself. Reliability of the extrapolations using thephotospheric magnetic field has to be further reviewed. Abbett(2007) demonstrated that the quiet Sun model chromosphere is anon-force-free layer, and the extrapolations from the upper chro-mosphere may accurately yield the quiet Sun field. Furthermore,our simulations do not capture the actual response of plasma tothe dissipation, which is the determining factor when comparingthe models with the emission maps of corona. Also, it is possiblethat the energy released during events such as magnetic recon-nection at some altitude can be dissipated at a different altitudedue to reconnection-driven processes, viz., jets, outflows, andwaves (Longcope 2004). These processes are not included in thepresent simulations. Thus a detailed MHD description is neces-sary to evaluate the role of the magnetic carpet in the heating ofthe solar atmosphere. To conclude, our results show that the en-ergy flux associated with quasi-static processes (here modeledusing magneto-friction and hyperdiffusion) are not sufficientto heat the quiet solar corona. Although this suggests that theevolution of the magnetic carpet (through magnetic splitting,merging, cancellation, and emergence), without invoking anyother reconnection-driven and wave mechanisms, does not playa dominant role in the coronal energy supply, a final judgmentcan be passed only with better observational constraints of mag-netic field, in particular, at the force-free upper chromosphericlayer. Observations from the next generation instruments suchas the Advanced Technology Solar Telescope and the Solar-C, which can offer multi-wavelength coverage with excellentspectro-polarimetric sensitivity and large FOVs, will shed lighton these issues.

The authors thank the referee for many comments andsuggestions that helped in improving the presentation of thework. L.P.C. was a 2011–2013 SAO Pre-Doctoral Fellowat the Harvard-Smithsonian Center for Astrophysics. Fund-ing for L.P.C. and E.E.D. is provided by NASA contractNNM07AB07C. Courtesy of NASA/SDO and the AIA andHMI science teams. This research has made use of NASA’sAstrophysics Data System.

APPENDIX

DETAILS OF THE SOURCE MODEL

In this Appendix, we consider a model for the evolution ofthe photospheric magnetic field of the quiet Sun. The field isassumed to consist of a collection of discrete flux elements or“sources.” Each source has an associated magnetic flux, whichcan be positive or negative, but the combined flux of all sources isassumed to vanish. Each source has a Gaussian flux distributionwith radius r0 = 200 km. The spatial distribution of thesources continually evolves due to several processes: (1) randommotions driven by sub-surface convective flows, (2) splittingand merging of like-polarity sources, (3) mutual cancellationof opposite-polarity sources, and (4) emergence of new bipoles.To simulate these processes, we introduce a hexagonal grid withan edge length L ≈ 1 Mm. Initially, the sources are centered atrandomly selected vertices of the hexagonal grid (where threeedges come together); subsequently, they start moving alongthe edges of the grid. Each source has an equal probability

0 10 20 30 40Time (hr)

0.0

0.5

1.0

1.5

Mag

netic

Flu

x (1

020 M

x)

Figure 10. Complete time evolution of the magnetic flux for the Source Modelintegrated over its FOV is shown. The plot includes both the rise and equilibriumphases of the magnetic flux as described in Section 2.1. Two thin vertical linesmark the period of the 8.5 hr duration from which the time sequence of theSource Model is extracted for further analysis.

(0.25) of moving along one of the three edges connecting toits original vertex or remaining fixed at that vertex. The motionoccurs with constant speed, v = 1.5 km s−1, so that at timet = Δt = L/v all moving sources again reach a neighboringvertex. Then the process repeats itself, producing a random walkof the sources along the grid edges. Therefore, in the presentmodel, all sources periodically return to the vertices (at timestn = nΔt with n = 1, 2, · · ·).

The magnetic sources interact with each other only at verticesof the hexagons. For example, when two sources meet at avertex, they are forced to merge or partially cancel each other,depending on their magnetic polarity. Also, each source has aprobability of splitting in two, which only occurs at a vertexso that the two parts may move away from each other alongdifferent edges. The emergence of a bipole is modeled byinserting a new pair of flux-balanced sources at a vertex andallowing the two sources to move apart along different edges ofthe grid. The newly inserted sources have absolute fluxes in therange 0.1–1.5 × 1018 Mx.

Figure 9 illustrates various interactions between sources:(a) flux emergence, (b) emergence and subsequent partialcancellation, (c) splitting of an element, and (d) total fluxcancellation. Each panel has two segments. The top segmentshows the time sequence of the respective interaction (thenumbers given in the snapshots indicate time, in minutes,elapsed since an arbitrary start time), and the bottom segmentshows the magnetic flux versus time. Flux merging and self-cancellation of a newly emerged bipole are other possible waysin which the elements interact. We emphasize that the motivationbehind the Source Model is not to create a realistic magneticflux distribution for the quiet Sun, but rather to have a modelwith an average flux density similar to that obtained in the IMaXobservations.

The simulation is initiated with 50 magnetic sources withmagnetic fluxes in the range of 5–50×1016 Mx. Figure 10 showsthe integrated flux of the sources (i.e., the sum of absolute valuesof fluxes) as function of time. The initial phase of the model isdominated by flux emergence. After 15 hr into the evolution,the integrated flux reached a statistical equilibrium of 1020 Mx.This is due to the balance between the new flux emergence andpartial/complete cancellation of the magnetic elements. The twothin vertical lines mark the period of evolution (8.5 hr) chosen

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for the main Source Model analysis presented in this work. Intotal, 2000 bipoles emerged in the complete time evolution ofthe model (360 during the 8.5 hr period). The average rate offlux emergence is about 150 Mx cm−2 day−1, which is a factorof three less than the value reported by Thornton & Parnell(2011). However, their study included a much wider range offlux emergence events (1016–1023 Mx).

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